The Structure of Cube Tilings Under Symmetry Conditions - Core

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Jul 19, 2012 - Zd and therefore the cluster tilings considered by them reduce to specific partitions of. Zd. Other ... For a given m, we define the (flat) torus Td.
Discrete Comput Geom (2012) 48:777–782 DOI 10.1007/s00454-012-9438-0

The Structure of Cube Tilings Under Symmetry Conditions Andrzej P. Kisielewicz · Krzysztof Przesławski

Received: 18 November 2011 / Revised: 27 April 2012 / Accepted: 2 July 2012 / Published online: 19 July 2012 © The Author(s) 2012. This article is published with open access at Springerlink.com

Abstract Let m1 , . . . , md be positive integers, and let G be a subgroup of Zd such that m1 Z × · · · × md Z ⊆ G. It is easily seen that if a unit cube tiling [0, 1)d + t, t ∈ T , of Rd is invariant under the action of G, then for every t ∈ T , the number |T ∩ (t + Zd ) ∩ [0, m1 ) × · · · × [0, md )| is divisible by |G|. We give sufficient conditions under which this number is divisible by a multiple of |G|. Moreover, a relation between this result and the Minkowski–Hajós theorem on lattice cube tilings is discussed. Keywords Cube tiling

1 Introduction An interest in cube tilings of Rd originated from the following question raised by Hermann Minkowski [29]: Characterize lattices Λ ⊂ Rd such that [0, 1)d + λ, λ ∈ Λ, is a cube tiling. Minkowski conjectured that such a lattice Λ is of the form AZd , where A is a lower triangular matrix for which every entry on the main diagonal equals 1. By a simple inductive argument it is equivalent to showing that Λ contains an element of the standard basis. Geometrically, it means that the tiling [0, 1)d + λ, λ ∈ Λ, contains a column. This inspired Ott-Heinrich Keller [16, 17] to consider the problem of the existence of columns in arbitrary (nonlattice) cube tilings. Eventually, Minkowski’s conjecture has been confirmed by Gy˝orgy Hajós [11], while Jeffrey Lagarias and Peter Shor [24] constructed a cube tiling without columns in dimension 10. Geometric A.P. Kisielewicz · K. Przesławski () Wydział Matematyki, Informatyki i Ekonometrii, Uniwersytet Zielonogórski, ul. Z. Szafrana 4a, 65-516 Zielona Góra, Poland e-mail: [email protected] A.P. Kisielewicz e-mail: [email protected]

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and algebraic questions stemming from these problems attracted quite a number of researchers; see, e.g., [2, 3, 5, 6, 18, 20, 25–28, 30, 35]. Numerous authors pay their attention to tilings by clusters of cubes; see, e.g., [10, 12, 13, 21, 31–34]. Their investigations are rooted in coding theory and the Golomb– Welch conjecture [9]. These authors often deal with cubes which have their corners in Zd and therefore the cluster tilings considered by them reduce to specific partitions of Zd . Other partitions of Zd called disjoint covering systems are discussed in number theory; see, e.g., [1, 7]. A new stimulus to the theory of tilings came from Fuglede’s conjecture [8]. A number of papers appeared at almost the same time where the set determining a cube tiling is characterized as follows: [0, 1)d + t, t ∈ T , is a cube tiling of Rd if and only if the system of functions exp(2π it, x), t ∈ T , is an orthonormal basis of L2 ([0, 1]d ) ([14, 15, 21, 23]). A reader who seeks an exposition concerning cube tilings is advised to consult [22, 33, 36]. In this work we shall be concerned with cube tilings of Rd that are invariant under the action of certain groups of translations. Let G ⊂ Rd be a group. A cube tiling [0, 1)d + t, t ∈ T , is G-invariant if T is G-invariant; that is, t + x ∈ T whenever t ∈ T and x ∈ G. Let us remark that if T is a lattice, then the cube tiling is T invariant. Observe that every G-invariant cube tiling leads to a tiling by clusters, where the union of all cubes [0, 1) + g, g ∈ G, serves as a prototile. Let m = (m1 , . . . , md ) be a vector whose coordinates are positive integers, and let e1 , . . . , ed be the standard basis of Rd . We say that the tiling [0, 1)d + t, t ∈ T , is m-periodic if it is invariant under the action of the group generated by the vectors mi ei , i = 1, . . . , d. (If m is unspecified, then we simply say that the tiling is periodic.) Our main result reads as follows. Theorem 1 Let m be a vector whose coordinates are positive integers, and G be a subgroup of Zd such that m1 Z × · · · × md Z ⊆ G. Let [0, 1)d + t, t ∈ T , be a G-invariant cube tiling of Rd . Let Gi = G ∩ Zei , and μi = |Z/Gi |. Then for every t ∈ T , the number |T ∩ (t + Zd ) ∩ [0, m1 ) × · · · × [0, md )| is divisible by GCD(μ1 , . . . , μd )|G|. We found it convenient to express our theorem in terms of cube tilings of flat tori. For a given m, we define the (flat) torus Tdm as the set [0, m1 ) × · · · × [0, md ) with addition mod m:   x ⊕ y := (x1 + y1 ) mod m1 , . . . , (xd + yd ) mod md . Cubes in Tdm are the sets of the form [0, 1)d ⊕t, where t ∈ Tdm . We say that T ⊂ Tdm determines a cube tiling of Tdm if the family T := {[0, 1)d ⊕ t : t ∈ T } is a tiling, that is, elements of T cover Tdm and are pairwise disjoint. Each m-periodic cube tiling of Rd defines a cube tiling of Tdm in an obvious manner. Those sets which determine cube tilings of Tdm are characterized as follows: Lemma 2 T determines a cube tiling of Tdm if and only if

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Fig. 1 The G-invariant cube tiling of T3(2,2,2) determined by T = {(0, 0, 0), (0, 0, 1), (1, 1, 0), (1, 1, 1), (1, 0, 1/4), (1, 0, 5/4), (0, 1, 1/4), (0, 1, 5/4)}. The group G consists of two elements, (0, 0, 0) and (1, 1, 1)

(1) |T | = m1 × · · · × md , (2) for every pair x, y ∈ Tm , there is i ∈ [d] such that xi − yi is a nonzero integer (Keller’s condition). (Compare [36, Lemma 8.3].) Let us mention that the notion of G-invariance carries over to cube tilings of Tdm . The integer lattice in Tdm is the subset Zdm := Zm1 × · · · × Zmd of Tdm consisting of all vectors with integer coordinates. If T determines a cube tiling T of Tdm and x ∈ T , then the family [0, 1)d ⊕ t, t ∈ T ∩ (x ⊕ Zdm ), is said to be a simple component of the tiling T . In light of Lemma 2, we shall be mostly concerned with sets that determine cube tilings rather than cube tilings themselves. Therefore, we shall refer to T ∩ (x ⊕ Zdm ) as a simple component of T . For u ∈ [0, 1)d , let Cu := T ∩ (u ⊕ Zdm ). It is clear that if Cu is nonempty, then it is a simple component of T . Moreover, for each simple component C of T , there is a unique element u ∈ [0, 1)d such that C = Cu . Theorem 1 can now be stated for the flat tori (Fig. 1). Theorem 3 Let T determine a cube tiling of Tdm . Let G be a subgroup of Zdm . Let Gi = G ∩ (Zmi ei ), where ei is the ith element of the standard basis, and μi = mi /|Gi |. If T is G-invariant, then each simple component of T has its cardinality divisible by GCD(μ1 , . . . , μd )|G|.

2 Proof of the Main Result We shall need the following: Lemma 4 Let V be a nonempty subset of [0, 1)d . If for every v ∈ V and every i ∈ [d], there is y ∈ V such that vq = yq whenever q = i and vi = yi , then for every v ∈ V , there is u ∈ V such that vr = ur whenever r ∈ [d].

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Proof By our assumptions, one can construct a sequence v j , j = 0, . . . , d, consisting j j −1 j j −1 of elements of V such that v 0 = v and vq = vq for q = j and vj = vj for j ∈ [d]. Now, it suffices to set u = v d .  Proof of Theorem 3 If the theorem were false, then there would exist a counterexample with minimal d. Clearly, d > 1, as for d = 1, the theorem is evidently true. Let n = GCD(μ1 , . . . , μd )|G|, and let V ⊂ [0, 1)d consist of all v such that Cv is a simple component of T and |Cv | is not divisible by n. As a hypothetical counterexample is under discussion, V is nonempty. We are going to show that V satisfies the assumptions of Lemma 4. We have to check only the case i = d, as for the remaining cases, the same reasoning applies. Let us define s : Tdm → Tdm by the formula s(t) = (t1 , . . . , td−1 , td ). Let S = s(T ). By Lemma 2, s restricted to T is a one-toone mapping, and S determines a cube tiling of Tdm . Moreover, since G is a subgroup of Zdm , it follows easily from the definition of S that S ⊕ G = {s ⊕ g : s ∈ S, g ∈ G} = S. For z ∈ [0, 1)d , let Bz = S ∩ (z ⊕ Zdm ). If x ∈ T ∩ [0, 1)d , then z = s(x) ∈ S ∩ [0, 1)d , and Bz is a simple component of S. Let m = (m1 , . . . , md−1 ). Then, by the definition of S, there are sets Si ⊂ Tm , i ∈ Zmd , such that S = S0 × {0} ∪ · · · ∪ Smd −1 × {md − 1}. Since S determines a cube tiling, each of the sets Si , i ∈ Zmd , determines a cube tiling of Tm . Let G be the subgroup of Tm defined by the equation G × {0} = {g ∈ G : gd = 0}. By the definition of Si and the fact that S ⊕ G = S we have Si ⊕ G = Si . Let G i = G ∩ (Zmi ei ) for i ∈ [d − 1]. Clearly, G i ∼ = Gi . Let z ∈ S ∩ [0, 1)d . The component Bz decomposes in a similar way to S: Bz = B0 × {0} ∪ · · · ∪ Bmd −1 × {md − 1}. If Bi is nonempty, then it is a simple component of Si . By induction, |Bi | is divisible by n = GCD(μ1 , . . . , μd−1 )|G | for every i ∈ Zmd . Let H = {gd : g ∈ G}. Since Bz is G-invariant, we have |Bi | = |Bj | for every pair i, j ∈ Zmd such that i − j ∈ H . Therefore, |Bz | is divisible by n |H | = GCD(μ1 , . . . , μd−1 )|G|. A fortiori, |Bz | is divisible by n. Let z = s(v), where v ∈ V , and let Ev = {w ∈ [0, 1)d ∩ T : s(w) = z}. Obviously Bz is a disjoint union of the sets s(Cw ), w ∈ Ev . Therefore, |Bz | =  w∈Ev |Cw |. Since v ∈ Ev and the number n divides |Bz |, and does not divide |Cv |, it follows that there is y ∈ Ev \ {v} such that n does not divide |Cy | as well. Thus, y belongs to V , and since s(v) = s(y), we have vq = yq whenever q = i(= d). It means that V satisfies the assumptions of Lemma 4 as expected. Consequently, there are two elements v and u in V such that vr = ur whenever r ∈ [d]. Let t ∈ Cv and t ∈ Cu . Then tj − tj is not an integer for any j ∈ [d], which violates Keller’s condition (Lemma 2). 

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3 Remarks A nonnegative integer n is representable by m ∈ Nd if there are nonnegative integers n1 , . . . , nd such that n = n1 m1 + · · · + nd md . We have proved the following theorem [19], appealing to the result of Coopersmith and Steinberger on cyclotomic arrays [4]: Theorem 5 If T determines a cube tiling of a torus Tdm and C is a simple component of T , then |C| is m-representable. It is a weakness of Theorem 3 that it is not a generalization of Theorem 5. The conjectural generalization should be the following: Let T determine a cube tiling of Tdm . Let G be a subgroup of Zdm . Let Gi = G ∩ (Zmi ei ), where ei is the ith element of the standard basis, and μi = mi /|Gi |. If T is G-invariant, then each simple component of T has the cardinality representable by (μ1 |G|, . . . , μd |G|). Such a generalization would lead, for example, to a new proof of the Minkowski– Hajós theorem on lattice cube tilings. We need less for this particular purpose. It would suffice to prove that, under the assumptions of Theorem 3, if all the numbers μ1 , . . . , μd are greater than 1, then at least one of the simple components of T has cardinality greater than |G|. Theorem 3, as it is, suffices to show a restricted p-adic version of the Minkowski–Hajós theorem: Let Λ ⊂ Rd be a cube tiling lattice. Suppose that there are a prime p and a positive integer k such that p k ei ∈ Λ for each i ∈ [d]. Then there is j ∈ [d] such that ej ∈ Λ. Indeed, let T = {λ mod m : λ ∈ Λ}, where m = (p k , . . . , p k ). By the fact that p k ei ∈ Λ for i ∈ [d], it follows that T determines a cube tiling of Tdm . Since Λ is a subgroup of Rd , T is a subgroup of Tdm . Let G = Zdm ∩ T . Then G is a nonempty subgroup of T , as 0 ∈ G. Therefore, T is G-invariant. By Theorem 3, the number |Zdm ∩ T | = |G| is divisible by GCD(μ1 , . . . , μd )|G|, which, by taking into account that all μi are powers of p, readily implies that there is j such that μj = 1. Thus, p k = |Gj |, and consequently, ej ∈ Gj ⊆ T . Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

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